Self-propelled particles align with neighbors: θᵢ(t+1) = ⟨θⱼ⟩_|r_ij|<r + η·ξ
Order param φ = |Σeⁱᶿ|/N—
N particles—
Avg neighbors—
Phase—
Vicsek model (1995): point particles move at constant speed, align with neighbors within radius r, plus angular noise η. Above a critical noise η_c: disordered phase (φ≈0). Below η_c: flocking (φ→1). This is a non-equilibrium phase transition with long-range order in 2D (violates Mermin-Wagner for equilibrium).